Lens-Like Beam-Forming Networks for Circular Arrays and Their Circuit Implementation

ABSTRACT

A beamforming apparatus comprising a number of beam ports N B  arranged in a circular array, the circular array having a radius a, an annular shaped lens encircling the number of beam ports N B , the annular shaped lens having an inner radius a, an outer radius b, and an inhomogeneous refractive index n(r), and a number of array ports N A  coupled to the outer rim of the annular shaped lens.

RELATED DOCUMENTS

The present application claims benefit under 35 U.S.C. §119(e) of U.S.Provisional Application No. 61/501,744 filed Jun. 27, 2011. Thisapplication is incorporated herein by reference in its entirety.

BACKGROUND

The present disclosure is related to the general field of antennasdesign, and particularly pertains to the subjects of electronicbeam-steering, feed network synthesis, and electromagneticmetamaterials. The present application describes a beamforming apparatusthat may be used for feeding circular ring arrays with 360° scancoverage, its basic design method, and its implementation as anelectrical network.

Beam steerable antennas with 360° scan coverage have long been a subjectof interest to the antenna community and have been extensively studiedfor the purpose of implementing radars and direction finders (DF). 360°beam-steering also has many potential applications in wirelesscommunication systems, for example when implementing base stationantennas for cellular communication or for wireless local area networks(WLAN).

360° beam steering can be implemented using circular ring antennaarrays, which in this case refers to an array of identical antennaelements arranged on a circle such that each element has an outwarddirected beam in a radial direction defined by the geometric center ofthe ring and the phase center of the element. In one implementation, asteerable beam is obtained by switching the electric signal between thedifferent antenna elements, or, in the case of direction finders, bydirectly reading the received signal at the output of individualelements. In this case, each element corresponds to a predefinedradiation pattern and beam.

Alternatively, the beam can be synthesized by combing the radiations (orreceived signals) from multiple array elements. In this case a largeportion of the array elements contribute to the transmitted or receivedsignal at any given time. Distributing the transmitted power between theradiating elements and combining the received signals is achieved by amulti-port electrical network called a “beam-forming network”, or BFN.

BFN is used to realize the desired transfer functions between theantenna elements and transmit/receive electronics. The ports at whichthe BFN couples to antennas are called “array ports”. The ports at whichthe BFN is coupled to the electronics are called beam port(s). BFN canbe fully passive and made up of fixed components, in which case it willbe used to generate a number of pre-determined radiation patterns(multi-beam antenna), or it may be active and include control devicesthat are used to set the amplitude and phase of excitation forindividual antenna elements on command, thereby providing anelectronically control radiation pattern and beam angle (phased array).In the contemporary language of antenna engineering, the termbeam-forming network usually refers to the first configuration and isused in the context of multi-beam antennas. The same meaning of the termis intended throughout this writing.

There are a number of designs for implementing the BFN for linearantenna arrays (arrays in which the antenna elements are arranged on astraight line), including “Butler matrix” (J. L. Butler and others “BeamForming Matrix Simplified Design of Electronically Scanned Antennas,”Electronic Design, pp 170-173, 1961) and “Rotman Lens” (W. Rotman andOthers, “Wide-Angle Microwave Lens for Line Source Applications,” IEEETransactions on Antennas and Propagation, pp. 623-632, 1963). In thesetypes of designs, the BFN can be implemented in the form of arectangular network, where the array ports and beam ports are built ontwo opposite sides of the rectangle. The phase delay relationshipsbetween the array ports and beams port are such that exciting the BFNfrom any given beam port creates the phase gradient necessary forproducing a beam in one of the desired directions. The power enteringBFN from each beam port distributes (nearly) equally between all antennaelements.

It is well understood that the most suitable geometry for the BFN in thecase of circular ring arrays with 360° scan is a circular geometry. Inthis case, the beamforming network plays two roles: 1) it produces thedesired phase relationship among array ports and 2) it concentrates thepower entering from each beam port predominantly among the antennaelements facing the direction of the desired output beam, that is, thosewhose maximum radiation occurs within ±90° of the output beam angle. Werefer to these elements as the “facing elements”. Designing a BFN thataccomplishes both of these tasks is difficult.

The design of circular BFN's has been addressed in the U.S. Pat. Nos.3,392,394, 5,274,389, and 3,754,270. All of these works rely ontwo-dimensional (2D) Luneberg lenses (G. D. M. Peeler and Others, “ATwo-Dimensional Microwave Luneberg Lens,” IRE Trans. Antennas andPropagation, pp. 12-23, 1953) or homogeneous approximation thereof toaccomplish the objectives of concentrating the power on the facing arrayports and synthesizing the required phase delays. However, due the factthat the radiating aperture and focal point of the Luneberg lens bothlie on its outer surface, in all of these designs the ports are definedon the rim of the lens double as both beam ports and array ports. Eachof these designs proposes a way for separating the beam ports and arrayports, but the resulting assembly is invariably cumbersome and for mostapplications undesirable.

The beamformer proposed in the U.S. Pat. No. 3,392,394 is implemented asa pair identical 2D Luneberg lenses each having an N number ports thatare stacked so that their ports line up on top of each other. Each ofthe N resulting double port stacks can be converted to a beam port andan array port by introducing a quadrature phase hybrid coupler. Thedrawback of this design is that it requires two Luneberg lenses and Nhybrid couplers that can occupy considerable space and introduce atleast some loss into the system.

A simpler version of the above design has been proposed in the U.S. Pat.No. 5,274,389 for a direction finder, where one of the Luneberg lensesis removed and the idle ports of the hybrid couplers is terminated bymatched resistors. This approach results in 6 dB loss in the signal thatis unacceptable in many applications.

The U.S. Pat. No. 3,754,270 proposes another beamformer that is based ona single N-port lens coupled to N isolators. If the isolators arecoupled to the lens ports at their first and are right handed (allowing1-2-3 rotation), in the transmit mode their second ports can act as thearray ports and their third ports as the beam ports. The presence of theisolators adds significant complexity to the design and is particularlyproblematic at higher frequencies. Also, for the same configuration towork on the receive mode, the sense of rotation in the isolators must bereversed (3-2-1 rotation). This requires the use electronicallycontrollable isolators (such as Farady rotation isolators (D. M. Pozar,Microwave Engineering, 3rd Edition: Wiley, 2004)) that furthercomplicates the implementation.

From these examples it is clear that the lack of separation between thearray ports and beam produces undesirable results. Introducingadditional building blocks to make this distinction leads to complexconfigurations that are often large, lossy, not realizable at highfrequencies, and not suitable for miniaturization and integration.

SUMMARY

Therefore, it is a first object of this specification to provide asolution for beamforming in circular ring arrays with 360° coverage thatcan be implemented using simple electrical components and with compactdimensions and provides a natural separation between the beam ports andarray ports. This beamforming apparatus: 1) directs the majority of thepower transmitted from each beam port towards the antenna elements thateffectively contribute to the radiation in the desired directionassociated with that beam port, 2) ensures that the radiations fromthese elements have the proper phase relationship, and 3) whoseimplementation does not require any switches or lossy elements. Abeamforming apparatus consisting of a nonhomogeneous annular lens deviceand a number of radially connected transmission line segments isproposed to address these features. We refer to this apparatus as the“Annular Lens Beam-Former” or ALBF.

A secondary object of this specification is to provide a method ofdetermining the value of the refractive index of the annular lens as afunction of the radius or n(r)—where r is the radius measured from thecenter of the lens—that accomplishes a first object of thisspecification. A design method based and formalism on geometrical opticsis provided to achieve this objective.

Another object of this specification is to provide a method forimplementing the annular lens of the secondary object as an electricalnetwork of capacitors, inductors, and transmission line segments. Adesign method and implementation based on techniques commonly used inthe field of metamaterials is introduced for deriving the circuitimplementation.

Yet another object of this specification is to describe how the proposedALBF can be used for the purpose of power combining.

An element of the proposed ALBF is a non-homogeneous annular lens, whichis designed using principles of geometrical optics. In an example ofthis device, the annular lens is implemented using 2D electromagneticmetamaterials in the form of a constrained electrical network with anannular topology, where the beam ports and array ports are defined bythe nodes lying on the inner and outer rims of the network,respectively. A signal launched from a beam port is primarilydistributed to the array ports that are located on the same side of theALBF (but on the outer rim), effectively exciting the portion of thecircular ring array that makes the greatest contribution to the farfield radiation in the direction of the beam. The annular lens alsoensures that signals reaching these elements have the proper phaserelationships used to synthesize a collimated output beam.

The annular topology of the ALBF physically separates the beam portsfrom the array ports and creates a natural isolation between the excitedbeam port and non-facing array ports, eliminating the need for hybridcouplers. Also the fact that annular lens is realized as an electricalnetwork with well-defined lumped ports eliminates the need for largetransitions and allows for very compact designs. The hollow space in themiddle of ALBF provides a natural place for accommodating transceiverelectronics and beam selection switches.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings illustrate various examples of the principlesdescribed herein and are a part of the specification. The illustratedexamples are merely examples and do not limit the scope of the claims.

FIG. 1 is a diagram of an electronically steerable antenna based on anannular lens beamformer (ALBF) device according to one example of theprinciples described herein.

FIG. 2 a visualizes the operation of an example multi-beam antennacomposed of a circular ring array of outward looking antenna elementsand an annular lens beamformer apparatus according to one aspect of thepresent specification. Facing [401] and non-facing [402] elements andelectromagnetic rays propagating inside the annular lens [501] andsurrounding medium [502] are shown for operation in the transmit modewith an output beam angle of φ^(b). The direction of the coordinatesystem axes and definitions of various angles are also shown.

FIG. 2 b visualizes the operation of an example multi-beam antennacomposed of a circular ring array of inward looking antenna elements andan annular lens beamformer apparatus according to another aspect of thepresent specification.

FIG. 3 is a diagram showing the electromagnetic rays between the beamport 1 and the facing array ports in an annular lens beamformer andtheir propagation in the surrounding medium according to one example ofthe principles described herein. For a properly designed lens, theelectromagnetic rays in the surrounding region will be parallel to eachother.

FIG. 4 shows the geometrical optics setup for calculating theelectromagnetic rays and designing the annular lens and defined variousgeometric parameters for a ray emanating from the first beam portlocated at r=a and φ=0.

FIG. 5 a. shows the calculated values refractive index for a multi-ringannular lens according to one example of the principles describedherein.

FIG. 5 b shows the calculated electromagnetic rays inside the annularlens according the example of FIG. 5 a.

FIG. 6 shows the array factor and radiation pattern calculated for thesemicircular ring array of the elements facing φ=0 direction and basedon the path lengths calculated for an example annular lens.

FIG. 7 a is a diagram showing the circuit topography and microstripimplementation of a two-dimensional positive-index metamaterialaccording to one example of the principles described herein.

FIG. 7 b is a diagram showing the circuit topography and microstripimplementation of a two-dimensional negative-index metamaterialsaccording to one example of the principles described herein.

FIG. 8 is an iso-frequency contour plot in the n_(x)-n_(y) plane for anegative-index metamaterial designed with the topology of FIG. 7 b,according to one example of the principles described herein.

FIG. 9 a is a square grid geometry for implementing a metamaterialannular lens beamformer according to one example of the principlesdescribed herein.

FIG. 9 b is a polar grid geometry for implementing a metamaterialannular lens beamformer according to one example of the principlesdescribed herein.

FIG. 10 depicts arrangement of the metamaterial unit cells andhomogeneous metamaterial rings for an annular lens implemented in polargrid metamaterials according to one example of the principles describedherein.

FIG. 11 a is a diagram depicting a single beam electronically steerableantenna configuration based on an annular lens beamformer according toone example of the principles described herein.

FIG. 11 b is a diagram depicting a multi-beam electronically steerableantenna configuration based on an annular lens beamformer according toone example of the principles described herein.

FIG. 11 c is a diagram depicting a radio relay antenna configurationbased on an annular lens beamformer according to one example of theprinciples described herein.

FIG. 11 d is a diagram depicting a retro-directive array configurationbased on an annular lens beamformer according to one example of theprinciples described herein.

FIG. 12 is a diagram showing an electronically-steerable antenna whichutilizes the annular lens beamformer described in this specification asfor beamforming and power combining network. Power amplifiers areintroduced between the beamformer array ports and antenna elements.

Throughout the drawings, identical reference numbers designate similar,but not necessarily identical, elements.

DETAILED DESCRIPTION

FIG. 1 shows a diagram of an electronically steerable antenna circularring array based on an annular lens beam-former (ALBF) device

according to one example of the principles described herein. In thistopology, ALBF is placed in between an N-element circular ring array[400] and the front-end electronic stage. An element of the ALBF is anannular lens like device [FIG. 2 a, 220] that is responsible forsynthesizing the proper phase and amplitude transfer functions forconstructing the desired array factor (beamforming). Antenna elements[FIGS. 1 and 2, 400 i, 401, 402] are coupled to this annular lens atN_(A) equally spaced “array ports” [FIG. 2 a, 231, 232], defined on theouter rim of the annular lens. The connection between the antennas andthe lens is generally through N_(A) equal-length transmission linesegments [300]. These transmission line segments do not alter therelative amplitudes and phases of ALBF transfer functions, but providean additional degree of freedom that allows the diameter of the annularlens [220] to be chosen independently of the diameter of the array. Thecoupling between the ALBF and front-end electronics is through N_(B)equally-spaced “beam ports” [FIG. 2 a, 211,212] that are defined on theinner rim of the annular lens. In general, N_(A) and N_(B) can bedifferent, but perfect rotational symmetry is attainable whereN_(A)=N_(B).

The annular lens beamformer is designed such that feeding from each beamport produces properly-phased excitations over the antenna elementsfacing the corresponding beam direction, that is, those antenna elementswhose direction of peak directivity lies within ±90° of the output beamangle. We refer to the elements fulfilling this condition as the “facingelements” [401]. The rest of the elements are referred to as the“non-facing elements” [402]. Similarly, the array ports connected to thefacing elements are referred to as the “facing array ports” [FIG. 2 a,231], and the array ports coupled to the non-facing elements arereferred to as “non-facing array ports” [FIG. 2 a, 232]. Consideringthat the direction of the beam is determined by the angle of the activebeam port [FIG. 2 a, 211], facing and non-facing elements and arrayports can be defined relevant to each beam port. The power radiated bythe non-facing elements is reduced due to the way that the wavepropagates inside the annular lens, and the residual radiations fromthese elements tend to cancel out in the far field as a result ofdestructive interference. A steerable or multi-beam radiation patternmay be produced using an RF switch [FIG. 1, 100] to select the desiredbeam port.

In the configuration of FIG. 1 and FIG. 2 a routing the signal to thebeam port located at angle φ^(B) results in an output beam in the samedirection [FIG. 2 a, 502]. FIG. 2 b shows an alternative topology inwhich the antenna elements have a backward radiation pattern. In thisconfiguration, to generate an output beam at angle φ^(B) the antenna isexcited from the beam port located at φ^(B)+π.

The refractive index of the annular lens is tailored so that acylindrical wave emanating from any of the beam ports excites theN_(A)/2 facing ports with the proper phase. In the present application,“proper phase” is meant to be understood as the phase needed for drivinga semicircular array of elements with radius c to produce a free-spacebeam at the angle φ^(B), where φ^(B) is as denoted in FIGS. 2 a and 2 b.The procedure for finding the appropriate refractive index function n(r)involves geometrical optics calculations and numerical optimizations.Due to the rotational symmetry, the problem is solved for one value ofφ^(B), most conveniently for φ^(B)=0.

The geometrical optics (GO) picture of the operating principle of theproposed beamforming apparatus is shown in FIG. 3. In this figure, theannular band occupying the region a≦r≦b represents the inhomogeneousannular lens. The point (r,φ)=(a,0) represents the excited beam port,and the facing array ports lie on the right half of the circle r=b. Thepoints marked by small circular bullets on the ring r=c denote the phasecenters of the antenna elements [403]. The region r>c represents thesurrounding propagation medium with refractive index n₀ (for antennasoperating in air or free space n₀=1). The bent lines connecting the beamport to each of the array ports [501] represent the optical paths (rays)inside the annular lens. The horizontal lines in the r>c region [502]represent the rays in the surrounding medium. The radial line connectingeach array port to the corresponding antenna element in the b<r<c region[503] represents propagation through the transmission lineinterconnecting the array port to the antenna element.

Assuming that the inhomogeneous annular lens is made up of M narrowconcentric rings (layers) with homogeneous refractive index n_(m), theelectromagnetic rays emerging from the beam port propagate on straightpaths inside each layer, but are refracted at the boundaries betweenadjacent layers according to the Snell's law of refraction. The opticalpath length inside each homogeneous region is defined as the product ofthe geometrical length of the ray in that region and the refractiveindex of the material filling that region. For the transmission linesegments the refractive index can be defined as ratio of the speed oflight (3×10⁸ m/s) to the phase velocity of the line, but in general theoptical length through the transmission line segments can be ignored,because it is the same for all array ports. The values of the inner andouter lens radii, a and b respectively, and the array radius c arechosen based on practical considerations and array requirements. n(r)(or the discrete values n₁ through n_(M)) is found such that all of theoutput rays reach a faraway point (r→∞) with the same optical pathlength. It can be shown that this is equivalent to the condition thatthe electromagnetic rays in the surrounding medium remain parallel tothe x axis.

FIG. 4 shows a detailed geometrical setup for calculating the opticalpath lengths and finding n(r). In this setup, the m'th homogeneous ringis defined by an inner radius r_(m), outer radius r_(m+1) and has therefractive index n_(m) (obviously r₁=a, r_(M+1)=b). The point from whichthe ray exists layer m is denoted by (r_(m+1), γ_(m)). The angle definedby the ray and the radial direction at its point of entry to the layeris denoted by α_(m) and angle between the ray and the radial directionat its point of exit from the layer is denoted by β_(m). The followingrelationships exist between these parameters:

$\begin{matrix}{{{\frac{\sin \; \beta_{m}}{r_{m}} = \frac{\sin \; \alpha_{m}}{r_{m + 1}}};{m = 1}},\ldots \mspace{14mu},M} & (1) \\{{{{n_{m}\sin \; \beta_{m}} = {n_{m + 1}\sin \; \alpha_{m + 1}}};{m = 1}},\ldots \mspace{14mu},{M - 1}} & (2) \\{{{\theta_{m} = {\sum\limits_{m^{\prime} = 1}^{m}( {\alpha_{m^{\prime}} - \beta_{m^{\prime}}} )}};{m = 1}},\ldots \mspace{14mu},M} & (3)\end{matrix}$

In order for the ray to enter the surrounding medium in parallel withthe x axis, one must have:

$\begin{matrix}{{n_{M}\sin \; \beta_{M}} = {{- \frac{c}{b}}n_{0}\sin \; \gamma_{M}}} & (4)\end{matrix}$

If this condition is satisfied, we will find:

$\begin{matrix}\{ {{\begin{matrix}{{\sin \; \alpha_{m}} = {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m}}\sin \; \gamma_{M}}} \\{{\sin \; \beta_{m}} = {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m + 1}}\sin \; \gamma_{M}}}\end{matrix};{m = 1}},\ldots \mspace{14mu},{M{and}\text{:}}}  & (5) \\{\gamma_{M} = {\sum\limits_{m = 1}^{M}\lbrack {{\sin^{- 1}( {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m}}\sin \; \gamma_{M}} )} - {\sin^{- 1}( {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m + 1}}\sin \; \gamma_{M}} )}} \rbrack}} & (6)\end{matrix}$

Theoretically, (6) must hold for all values of γ_(M) in the [−π/2, π/2]interval or thanks to its symmetry for [0, π/2]. However, due to thefinite number of rings in the array this condition cannot be metperfectly for the entirety of this interval.

A first method of designing the annular lens is, therefore, to enforce(6) for M values of γ_(M), preferably chosen as φ₁ ^(A), φ₂ ^(A), . . ., φ_(M) ^(A), the angles of the M array ports in the upper rightquadrant of the ALBF closest to the x axis, and solve the resultingsystem of nonlinear equations for n₁ through n_(M).

For a ray exiting at angle γ_(M) the optical length can be calculatedas:

$\begin{matrix}{{L( \gamma_{M} )} = {{\sum\limits_{m = 1}^{M}{n_{m}r_{m}\frac{\sin ( {\alpha_{m} - \beta_{m}} )}{\sin \; \beta_{m}}}} = {\sum\limits_{m = 1}^{M}{\frac{n_{m}^{2}r_{m}r_{m + 1}}{n_{0}c}{\sin \lbrack {{\sin^{- 1}( {\frac{n_{0}c}{n_{m}r_{m}}\sin \; \gamma_{M}} )} - {\sin^{- 1}( {\frac{n_{0}c}{n_{m}r_{m + 1}}\sin \; \gamma_{M}} )}} \rbrack}}}}} & (7)\end{matrix}$

The phase of the signal reaching an array port at angle φ_(i) ^(A) canbe calculate from the optical path length for L(φ_(i) ^(A)). Assumingthe excited ports all have the same amplitude, one can use this tocalculate the output array factor as:

$\begin{matrix}{{{AF}( {\theta,\phi} )} = {\sum\limits_{i}{\exp \lbrack {{{- j}\; k_{0}{L( \phi_{i}^{A} )}} + {j\; k_{0}n_{0}c\; \sin \; \theta \; {\cos ( {\phi - \phi_{i}^{A}} )}}} \rbrack}}} & (8)\end{matrix}$

where θ and φ are the polar angles in the direction of observation, k₀is the free space wave number at the design frequency, and the summationis over the N_(A)/2 facing array ports. A second method of designing thelens is, therefore, to find the refractive indexes n₁ through n_(M) thatmaximize the magnitude of |AF(π/2,0)|. This can be set up and solved asan optimization problem using any of the variety of known nonlinearoptimization techniques. Under ideal circumstances, the optimal value of|AF(π/2,0)| approaches N_(A)/2.

The calculated value of n(r) for a 5-ring 16-output ALBF for a certainchoice of a, b, and c is shown in FIG. 5 a. In this example the lens wasdivided into 5 equal width rings and the second method described abovewas used to find the values of n. Computed electromagnetic rays areshown in FIG. 5 b. FIG. 6 shows the normalized array factor of the32-element ring array when driven from the port at φ=0. The overallradiation pattern calculated by assuming a cardioid element pattern ofcos²(φ/2−φ_(i) ^(A)/2) for the element located at (c, φ_(i) ^(A)) isalso shown.

As seen from this example, the refractive index n(r) is negative. Thisis indeed always true if the ALBF is to be used with forward radiatingantenna elements (configuration FIG. 2 a), and can be explained by thefact that the array ports fed the elements that are father from theobservation point are fed corresponds to the rays with greatergeometrical length inside the annular lens. Therefore, we refer to thislens as the negative index ALBF, or simply “N-ALBF”. Negative refractiveindex is not observed in natural materials at microwave frequencies, butit can be realized in narrow bandwidths using metamaterials.

If the radiating elements in the circular array have a backward pointingradiation pattern, as in the configuration of FIG. 2 b, a similarprocess can be used to derive an lens beamformer design that whenexcited from an electronic port located at angle φ_(i) produces a beamin the direction φ_(i)+π. According to the second design methoddescribed above, this requires |AF| of (8) to be maximized for(θ,φ)=(π/2, π). It can be observed that the values of n(r) in this casewill be positive and equal to the absolute value of those found for theN-ALBF. This type of beamformer, which we refer to as the positive-indexALBF or simply “P-ALBF”, can be realized using natural materials as wellas low-pass artificial dielectrics and positive-index metamaterials andtherefore is conducive to wideband true-time-delay type implementations.In practice, to avoid blockage and cross talk between elements, P-ALBFmay be used with antenna elements with backward-tilted radiation patternrather than those with backward in-plane beams. Such an assembly will besuitable for generating a scanning beam above or below—but not in—theALBF plane. Consequently, the array factor is optimized for (θ, φ)=(θ₀,π) where θ₀ is the actual elevation angle of the desired beam. It can beshown that this is equivalent to replacing c with −c.sin θ₀ in equations(4)-(8).

Although it may be possible to implement the annular lens separately andcouple it to the beam ports and array ports through properly designedtapered transitions, in one example of ALBF, the annular lens isimplemented as an electrical network of inductors, capacitors, andtransmission line segments. In a network implementation of this kind,beam ports and array ports can be directly defined between the properlychosen terminals of the network, hence eliminating the need fortransitions. This presents an advantage over formerly introducedlens-based beamformers, and allows for unprecedented levels ofminiaturization and integration.

A network implementation of ALBF can also be obtained by using ametamaterial approach. The metamaterial formulation in this case can beviewed as a wave-based network design approach that provides approximaterecursive relationships between the nodal voltages and currents in everysmall region of the network by assuming that the local properties aresimilar to those of an infinite lattice. As these properties are easilytranslatable to an effective refractive index, a metamaterial outlook isideal for deriving network implementations of the optically designedALBF.

Turning now to FIGS. 7 a and 7 b, two diagrams are shown depictingexample circuit topographies and two-dimensional (2D) microstripimplementations of a positive-index metamaterial and a negative-indexmetamaterial, respectively, according to one example of the principlesdescribed herein. Although these implementations are distributed andexhibit higher order resonances, the useful band of operation fortopology 7 a is from dc to nearly half of its Bragg frequency and fortopology 7 b is the lower portion of the fundamental pass-band.Generally speaking, positive index metamaterials can be implemented inboth low-pass and band-pass topologies, while negative indexmetamaterials are invariably band-pass. In the microstrip implementationproposed in FIGS. 7 a and 7 b, capacitors and inductors are implementedin the form of planar capacitive gaps and vias, respectively. Thesechoices of positive and negative metamaterial are merely examples thathave been included for the purpose of illustration. In practice, otherimplementations may be needed to realize the desired values ofcapacitance and inductance. Other cell implementations and grid types(e.g. triangular or hexagonal) can also be used to implement the annularlens. The topology and design techniques for implementing 2Dpositive-index-of-refraction and negative-index metamaterials have beenwidely discussed in the literature known to those with the knowledge ofthe field (F. Capolino, Metamaterials Handbook CRC Press, 2009; C. Calozand T. Itoh, Electromagnetic metamaterials: transmission line theory andmicrowave applications: the engineering approach. Hoboken, N.J.: JohnWiley & Sons, 2006.).

Dispersion relations for metamaterials in infinite lattice are obtainedby solving the circuit equations for the each unit cell subject toproper periodic boundary conditions. These relationships give theper-unit-cell phase delays along each of the principal axes in terms offrequency and direction of propagation, which in the case oftwo-dimensional metamaterials, can be concisely presented byiso-frequency contour plots in the k_(x)d-k_(y)d plane (where d is thegrid constant). A useful presentation for the purposes of the presentapplication is one of iso-frequency contour plots in the n_(x)-n_(y)plane, where n_(x)=k_(x)/k₀ and n_(y)=k_(y)/k₀. A sample plot obtainedfor a negative-index metamaterial design with the topology of FIG. 7 bis shown in FIG. 8, where the contours have been calculated forfrequencies in the lower half of the lattice pass-band (below cellresonant frequency).

It has been stated that, if the unit cell dimensions are smaller than1/10 of the operating wavelength, metamaterials can be treated asnatural materials in the sense that they can be cut into arbitraryshapes and yet maintain their infinite grid properties (C. Caloz and T.Itoh, Electromagnetic metamaterials: transmission line theory andmicrowave applications: the engineering approach. Hoboken, N.J.: JohnWiley & Sons, 2006). This condition is commonly known as the “conditionof homogeneity”. Meeting this condition is difficult at upper microwaveand millimeter-wave bands due to the manufacturing limitations.Fortunately, the condition of homogeneity can be dispensed with as longas: (1) the operation frequency is within the lowest pass-band of themetamaterial, (2) the sample geometry can be made up of complete unitcells, and (3) the termination conditions for the boundary elements aretaken into account. One practical limitation in the case of annular lensis that the circular boundaries that define the homogeneous rings of thelens do not conform to common regular grids and pixelation errors canbecome significant for practical unit cell dimensions. This effect hasbeen illustrated in FIG. 9 a. Also, if the annular lens is implementedusing a standard grid topology, ALBF will not be rotationally symmetricand the beam ports and array ports cannot be placed at equal angularintervals, which is necessary for maintaining a consistent radiationpattern and port impedance.

The above limitations can be overcome by using a polar grid, as shown inFIG. 9 b. The advantage of the proposed polar grid topology is that itperfectly conforms to the contours of the ALBF, and clearly preservesthe rotational symmetry. As shown in FIG. 10, if sectoral divisions aresufficiently small and the radial grid constant is scaled proportionallywith r so that the grid pitch is nearly the same in the radial andazimuth directions, each cell in the polar grid can be approximated by acell in an equivalent square lattice of the same dimensions. Thanks tothis approximation, each ring in the polar grid metamaterial can bedesigned exactly in the same manner as a regular lattice with similargrid constant. The values of the outer and inner radii and number ofgrid sectors in the φ direction are chosen based on the practical rangeof capacitance and inductance values available in the technology ofchoice as well as the self-resonance of the unloaded cell. Theseparameters are chosen such that the cell dimensions for outermost ringdo not exceed one half of a guided wavelength (for the transmissionlines that are used to construct the cells). Theoretically, the bestresult is obtained if the each ring of the polar grid metamaterialconforms with a layer of the optically designed annular lens. Therefore,the task of ALBF design may start from choosing the parameters of thepolar grid and using the resulting values of cell radii to discretize rfor the purpose of designing the optical lens.

It must be understood that a metamaterial formulation and designapproach is only one way of deriving a circuit implementation for theALBF. Once the circuit topology of the lens and unit cells has beendetermined according to the principles described herein, it is easy fora person with ordinary skills in the art to devise other direct circuitsynthesis methods or optimization techniques to determine the values ofinductance, capacitors and transmission line components used toconstruct the lens.

An observation is that neither the optical design of the annular lensnor the metamaterial design of the ALBF electrical network presentedabove consider the effects of impedance mismatch between the lenslayers. Also, the effects of reflection from the outer boundaries of theALBF or the beam ports and array ports are not included in the designprocess. These effects can be addressed a posteriori by adding properlydesigned matching networks to the beam ports and array ports.

The application of the proposed annular lens beamformer devices is notlimited to single beam electronically steerable antennas. Some of theuseful configurations are shown in FIGS. 11 a through 11 d. While thebasic ESA configuration of FIG. 11 a, is suitable for a mobile terminalcommunicating with a fixed base station or another mobile terminal, aconfiguration with multiple independently controlled beams such as thatdepicted in FIG. 11 b can be ideal for a hub simultaneouslycommunicating with two or more mobile users. The configuration of FIG.11 c depicts an ALBF based antenna operating as a relay station(repeater). Finally, the configuration as depicted in FIG. 11 d is aretrodirective array and appropriate for communication between a mobilenode and a remote interrogator.

In high power applications, such as for cellular base stations, ALBF mayalso be integrated with power amplifiers to merge the functions ofbeamforming and power-combining in one device. An example of such aconfiguration is shown in FIG. 12, where power amplifiers are addedbetween the array ports and the antenna elements.

1. A beamforming apparatus comprising: a number of beam ports N_(B)arranged in a circular array, the circular array having a radius a; anannular shaped lens encircling the number of beam ports N_(B), theannular shaped lens having an inner radius a, an outer radius b, and aninhomogeneous refractive index n(r); and a number of array ports N_(A)coupled to the outer rim of the annular shaped lens.
 2. The apparatus ofclaim 1 in which: the number of beam ports N_(B) are equally spacedalong the internal rim of the lens; the number of array ports N_(A) areequally spaced along the external rim of the lens; the annular shapedlens is positioned in the xy plane such that its center is located atthe origin of the xy plane and a first beam port of the number of beamports N_(B) aligns with the positive x axis; the number of beam portsN_(B) are labeled from 1 through N_(B) starting from the first beam portand continuing in a consecutive counter clockwise order; the number ofarray ports N_(A) are labeled from 1 through N_(A), starting from thearray port closest to the x axis and in the first (upper right) quadrantof the xy plane and continuing in a consecutive counter clockwise order;the number of array ports N_(A) are positioned symmetrically withrespect to the x axis; the position of the number of array ports N_(A)with respect to the x axis is denoted by angles φ_(i) ^(A); and theposition of the number of beam ports N_(B) with respect to the x axis isdenoted by angles φ_(i) ^(B).
 3. The apparatus of claim 2, in which:each of the number of array ports N_(A) are connected to a first end ofa transmission line segment of a number of equally long transmissionline segments; a number of antenna elements are each connected to asecond end of the number of transmission line segments in which: a. thenumber of antenna elements N_(A) have phase centers that are arranged ina circular array of radius c; and b. the maximum directivity of each ofthe number of antenna elements occurs in the xy plane and in the outwarddirection defined by a vector with a first point at the geometric centerof the annular shaped lens and a second point at the phase center of theantenna element.
 4. The apparatus of claim 3, in which the annularshaped lens comprises a number of concentric rings M having homogeneousrefractive indexes n₁ through n_(m).
 5. The apparatus of claim 4, inwhich: the refractive indexes n₁ through n_(M) of the number ofconcentric rings are found by solving the following system of equations:${{\phi_{i} = {\sum\limits_{m = 1}^{M}\lbrack {{\sin^{- 1}( {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m}}\sin \; \phi_{i}} )} - {\sin^{- 1}( {{- \frac{n_{0}}{n_{m}}}\frac{c}{r_{m + 1}}\sin \; \phi_{i}} )}} \rbrack}};{i = 1}},\ldots \mspace{14mu},M$in which: a. n₀ is the refractive index of the propagation mediumsurrounding the antennas; b. r₁ through r_(M) are the inner radii of theconcentric rings; c. r₂ through r_(M+1) are the outer radii of theconcentric rings; d. φ₁, φ₂, . . . , φ_(M) are M distinct arbitraryvalues in the range of (0, π/2).
 6. The apparatus of claim 5, in whichM≧N_(A)/4 and φ₁, φ₂, . . . , φ_(M) are chosen as φ₁ ^(A), φ₂ ^(A), . .. , φ_(M) ^(A) or the angles of array ports 1 through M with respect tothe x axis.
 7. The apparatus of claim 4, in which the refractive indexesn₁ through n_(M) are found from maximizing the following array factor orany equivalent thereof:${{AF}} = {{\sum\limits_{i}{\exp\lbrack {{{- j}\; k_{0}{\sum\limits_{m = 1}^{M}\{ {\frac{n_{m}^{2}r_{m}r_{m + 1}}{n_{0}c}{\sin \lbrack {{\sin^{- 1}( {\frac{n_{0}c}{n_{m}r_{m}}\sin \; \phi_{i}^{A}} )} - {\sin^{- 1}( {\frac{n_{0}c}{n_{m}r_{m + 1}}\sin \; \phi_{i}^{A}} )}} \rbrack}} \}}} + {j\; k_{0}n_{0}c\; \cos \; \phi_{i}^{A}}} \rbrack}}}$in which: a. k₀ is the wave number in the surrounding medium at thedesign frequency; and b. the outer summation is over all or a subset ofthe array ports which lie on the right half of the xy plane (x>0). 8.The apparatus of claim 4, in which the refractive index values n₁through n_(M) are found such that the optical path lengths from eachbeam port with an angle φ_(i) ^(B) through any of the array ports withangles φ_(i) ^(A) fulfilling the condition |φ_(i) ^(A)−φ_(i) ^(B)≦π2 toan infinitely far away point on the φ=φ_(i) ^(B) line are equal.
 9. Theapparatus of claim 3, in which the refractive index n(r) is found suchthat the optical path lengths from each beam port with an angle φ_(i)^(B) through any of the array ports with angles φ_(i) ^(A) fulfillingthe condition |φ_(i) ^(A)−φ_(i) ^(B)≦π/2 to an infinitely far away pointon the φ=φ_(i) ^(B) line are equal.
 10. The apparatus of claim 4, inwhich the annular lens comprises a number of two-dimensionalnegative-index metamaterials in which the metamaterial unit cells: areimplemented using a negative-index topologies; and are connectedtogether in a periodic grid configurations.
 11. The apparatus of claim10, in which metamaterial unit cells are composed of lumped anddistributed components.
 12. The apparatus of claim 11, in which theannular shaped lens is implemented using polar grid metamaterials, inwhich: a number of P×Q grid points are found by intersecting a first setof P concentric circles and a second set of Q equally spaced radiallines; each metamaterial unit cell is centered at one of the number ofP×Q grid points; unit cells centered on each of the P concentric circlesare identical to each other and form a homogeneous metamaterial ring;and the unit cells lying on different circles generally differ in size,component values, and refractive index.
 13. The apparatus of claim 12,in which: P is greater or equal to the number of homogeneousmetamaterial rings in the annular lens; Q is an integer multiple of bothN_(A) and N_(B); each homogeneous metamaterial ring of the lens isimplemented using a one or an integer number of the metamaterial rings;the number of beam ports are implemented by periodically tapping intothe inner most metamaterial ring by making parallel or seriesconnections with or otherwise coupling to the nodes or branches of theouter metamaterial ring; and the number of array ports are implementedby periodically tapping into the outer most metamaterial ring by makingparallel or series connections with or otherwise coupling to the nodesor branches of the outer metamaterial ring.
 14. The apparatus of claim13, in which: the separation between successive P concentric circles andthe width of the metamaterial rings grows proportionally with theirradius; the separation between successive P concentric circles is nearlyequal to the separation between adjacent radial grid line at any givenradius; the length of each metamaterial unit cell, as measured along theradial direction, is nearly equal to its average width, as measured inthe azimuth direction; and the effective properties of each unit cellcan be approximated by those of a metamaterial unit cell implemented ina regular square grid of the same cell length.
 15. The apparatus ofclaim 13, in which: the separation between successive P concentriccircles and the width of the metamaterial rings are chosen arbitrarily;the length of the metamaterial unit cell, as measured along the radialdirection, may be smaller or greater than to its average width, asmeasured in the azimuth direction; and the effective properties of eachunit cell can be approximated by those of a metamaterial unit cellimplemented in a regular rectangular grid of the same cell length andwidth.
 16. The apparatus of claim 3, in which the annular lens isimplemented as an electrical network of interconnected bandpass buildingblocks each comprising capacitors, inductors, and transmission lines andhaving a center frequency greater than the intended operation frequencyof the lens and a lower cutoff frequency smaller than the intendedoperation frequency of the lens.
 17. The apparatus of claim 3, in whichimpedance matching networks are added to the number of beam ports N_(B)and number of array ports N_(A).
 18. The apparatus of claim 2, in whicha maximum directivity of each of the number elements occurs: at theelevation angle θ=θ₀ where θ is the elevation angle measured in thepolar coordinate system defined by the xy plane and θ₀ is the desiredelevation angle of the array beam; and at the azimuth direction definedby an inward radial vector with a first point at the phase center of theantenna element and a second point at the geometric center of theannular shaped lens.
 19. The apparatus of claim 18, in which the annularlens comprises a number of concentric rings M of homogeneous refractiveindexes n₁ through n_(M).
 20. The apparatus of claim 19, in which therefractive indexes n₁ through n_(M) are found from solving the followingsystem of equations or an equivalent thereof:${{\phi_{i} = {\sum\limits_{m = 1}^{M}\lbrack {{\sin^{- 1}( {\frac{n_{0}}{n_{m}}\frac{d}{r_{m}}\sin \; \phi_{i}} )} - {\sin^{- 1}( {\frac{n_{0}}{n_{m}}\frac{d}{r_{m + 1}}\sin \; \phi_{i}} )}} \rbrack}};{i = 1}},\ldots \mspace{14mu},M$in which d=c cos θ₀ and: a. n₀ is the refractive index of thesurrounding propagation medium; b. r₁ through r_(M) are the inner radiiof the concentric rings; c. r₂ through r_(M+1) are the outer radii ofthe concentric rings; d. φ₁, φ₂, . . . , φ_(M) are M distinct arbitraryvalues in the range of (0, π/2).
 21. The apparatus of claim 20, in whichφ₁, φ₂, . . . , φ_(M) are chosen as φ₁ ^(A), φ₂ ^(A), . . . , φ_(M) ^(A)or the angles of array ports 1 through M with respect to the x axis. 22.The apparatus of claim 19, in which the refractive indexes n₁ throughn_(M) are found from maximizing the following array factor or anyequivalent thereof:${{AF}} = {{\sum\limits_{i}{\exp\lbrack {{{- j}\; k_{0}{\sum\limits_{m = 1}^{M}\{ {\frac{n_{m}^{2}r_{m}r_{m + 1}}{n_{0}c}{\sin \lbrack {{\sin^{- 1}( {\frac{n_{0}d}{n_{m}r_{m}}\sin \; \phi_{i}^{A}} )} - {\sin^{- 1}( {\frac{n_{0}d}{n_{m}r_{m + 1}}\sin \; \phi_{i}^{A}} )}} \rbrack}} \}}} + {j\; k_{0}n_{0}d\; \cos \; \phi_{i}^{A}}} \rbrack}}}$in which: a. d=c cos θ₀; b. k₀ is the free space wave number at thedesign frequency; and c. the outer summation is over all or a subset ofthe array ports which lie on the right half of the xy plane (x>0). 23.The apparatus of claim 19, in which the refractive index values n₁through n_(M) are found using any other method so that that the opticalpath lengths from each beam port with an angle φ_(i) ^(B) through any ofthe array ports with angles φ_(i) ^(A) fulfilling the condition |φ_(i)^(A)−φ_(i) ^(B)|≦π/2 to an infinitely far away point on the φ=φ_(i)^(B)±π line are equal.
 24. The apparatus of claim 18, in which therefractive index n(r) is found so that the optical path lengths fromeach beam port with an angle φ_(i) ^(B) through any of the array portswith angles φ_(i) ^(A) fulfilling the condition |φ_(i) ^(A)−φ_(i)^(B)|≦π/2 to an infinitely far away point on the φ=φ_(i) ^(B)±π line areequal.
 25. The apparatus of claim 19, in which the annular lens isrealized using two-dimensional positive-index metamaterials in which themetamaterial unit cells: are implemented using positive-indextopologies; and are connected together in a periodic gridconfigurations.
 26. The apparatus of claim 25, in which the metamaterialunit cells are composed of lumped and distributed components.
 27. Theannular lens of claim 26, in which the annular shaped lens isimplemented using polar grid metamaterials, in which: a number of P×Qgrid points are found by intersecting a first set of P concentriccircles and a second set of Q equally spaced radial lines; eachmetamaterial unit cell is centered at one of the number of P×Q gridpoints; unit cells centered on each of the P concentric circles areidentical to each other and form a homogeneous metamaterial ring; unitcells lying on different circles generally differ in size, componentvalues, and refractive index; in which: P is greater or equal to thenumber of homogeneous metamaterial rings in the annular lens; Q is aninteger multiple of both N_(A) and N_(B); each homogeneous metamaterialring of the lens is implemented using one or an integer number ofmetamaterial rings; the number of beam ports are implemented byperiodically tapping into the inner most metamaterial ring by makingparallel or series connections with or otherwise coupling to the nodesor branches of the inner most metamaterial ring; and the number of arrayports are implemented by periodically tapping into the outer mostmetamaterial ring by making parallel or series connections with orotherwise coupling to the nodes or branches of the outer mostmetamaterial ring, and the separation between successive P concentriccircles and the width of the metamaterial rings grows proportionallywith their radius;
 28. The apparatus of claim 27, in which: theseparation between successive P concentric circles is nearly equal tothe separation between adjacent radial grid line at any given radius;the length of each metamaterial unit cell, as measured along the radialdirection, is nearly equal to its average width, as measured in theazimuth direction; and the effective properties of each unit cell can beapproximated by those of a metamaterial unit cell implemented in aregular square grid of the same cell length.
 29. The apparatus of claim27, in which: the separation between successive P concentric circles andthe width of the metamaterial rings are chosen arbitrarily; the lengthof the metamaterial unit cell, as measured along the radial direction,may be smaller or greater than to its average width, as measured in theazimuth direction; and the effective properties of each unit cell can beapproximated by those of a metamaterial unit cell implemented in aregular rectangular grid of the same cell length and width.
 30. Theapparatus of claim 18, in which the annular lens is implemented as anelectrical network of interconnected low-pass building blocks eachcomprising capacitors, inductors, and transmission lines and having acutoff frequency greater than the intended operation frequency of thelens.
 31. The apparatus of claim 18, in which the annular lens isimplemented as an electrical network of interconnected bandpass buildingblocks each comprising capacitors, inductors, and transmission lines andhaving a center frequency smaller than the intended operation frequencyof the lens and an upper cutoff frequency greater than the intendedoperation frequency of the lens.
 32. The apparatus of claim 18, in whichimpedance matching networks are added to the number of beam ports andthe number of array ports.
 33. A beamforming apparatus comprising: anumber of beam ports N_(B) arranged in a circular array, the circulararray having a radius a; an annular shaped lens encircling the numberbeam ports N_(B), the annular shaped lens having an inner radius a, anouter radius b, and an inhomogeneous refractive index n(r); and a numberof array ports N_(A) coupled to the outer rim of the annular shapedlens; in which a number of power amplifiers are added to the arrayports.
 34. A beamforming apparatus comprising: a number of beam portsN_(B) arranged in a circular array, the circular array having a radiusa; an annular shaped lens encircling the number beam ports N_(B), theannular shaped lens having an inner radius a, an outer radius b, and aninhomogeneous refractive index n(r); and a number of array ports N_(A)coupled to the outer rim of the annular shaped lens; in which: thenumber of beam ports N_(B) are equally spaced along the internal rim ofthe annular shaped lens; the number of array ports N_(A) are equallyspaced along the external rim of the annular shaped lens; each of thenumber of array ports N_(A) are connected to a first end of atransmission line segment of a number of equally long transmission linesegments; a number of antenna elements are each connected to a secondend of the number of transmission line segments in which: a. the numberof antenna elements have phase centers that are arranged in a circulararray of radius c; and b. a maximum directivity of each of the number ofantenna elements occurs in the xy plane and in the outward directiondefined by a vector with a first point at the geometric center of theannular shaped lens and a second point at the phase center of theantenna element the annular shaped lens comprises a number of concentricrings M of homogeneous refractive indexes n₁ through n_(M); and a numberof power amplifiers are added in between a pair of successivemetamaterial rings.